mt185-sample1 - z 7 (1 + i ) z + 1. (b). Describe the image...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 185 Sample First Midterm [This may be a bit longer than our exam will be.] 1. (20 points) Carefully define the following. (In each definition you may use without defining them any terms or symbols that were used in the text prior to that definition.) (a). Analytic function (b). Closed set (c). ± -neighborhood (d). Branch point (e). z c ( z,c C , z 6 = 0) 2. (10 points) Prove that if f 0 ( z ) = 0 everywhere in a domain D , then f ( z ) must be constant throughout D . In proving this, you should be careful not to use any results from later in the book that rely on this result. 3. (10 points) Find all complex solutions to the equation z 6 - 1 2 + i 2 = 0. You may use polar coordinates. 4. (15 points) Let S = { z C : 0 Re z 1, 0 Im z 2 π } . (a). Describe the image of S under the map
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: z 7 (1 + i ) z + 1. (b). Describe the image of S under the map z 7 e z . (c). Describe the image of S under the map z 7 Arg( z + i ). (Arg denotes the principal branch of the argument.) (d). Describe the images of S under the maps z 7 Arg ( z + i ), where Arg is any branch of the argument dened on S . 5. (15 points) Let f ( z ) = x 3 + i (1-y ) 3 (where z = x + iy as usual). Show that the only point where f is dierentiable is z = i . 6. (15 points) Let f ( z ) = f ( x + iy ) = u ( x,y )+ iv ( x,y ) is entire and that the rst-order partial derivatives of u and v are continuous. Let h ( z ) = f ( z ). Show that h ( z ) is entire....
View Full Document

Ask a homework question - tutors are online